Data display - Practice

What you will learn

distinguish categorical from numerical data, and discrete from continuous data,
build a frequency table,
construct and read column graphs, dot plots, stem-and-leaf plots, and line graphs,
choose the best display for a given data set.

1. Types of data

Data types

Categorical

Data that sorts into groups or categories. Examples: eye colour, favourite subject, yes/no.

Numerical - discrete

Counting data - separate whole values. Examples: number of siblings, number of goals.

Numerical - continuous

Measuring data - can take any value in a range. Examples: height, time, weight.

2. Frequency tables

A frequency is the count of how many times a value or category appears.

Worked example 1 Frequency table

Twenty students were asked their favourite colour. Build a frequency table from this list:

Red, Blue, Red, Green, Red, Blue, Yellow, Red, Blue, Green, Red, Red, Blue, Yellow, Green, Red, Blue, Red, Green, Blue.

Colour	Frequency
Red	$7$
Blue	$6$
Green	$4$
Yellow	$2$
Other	$1$

Check: $7 + 6 + 4 + 2 + 1 = 20$ .

As a column graph:

Column graph of favourite colours. Each bar's height matches the frequency in the table above. Gaps between bars show this is categorical data.

3. Types of graph

Choosing a display

Column / bar graph

Use for categorical data or discrete numerical data. Bars have gaps between them.

Dot plot

Use for small discrete numerical data sets. Each dot represents one data value stacked above its number on the axis.

Stem-and-leaf plot

Use for numerical data when you want to keep individual values. The stem is the leading digit(s), the leaf is the final digit.

Line graph

Use when data is continuous over time, e.g. temperature during a day.

Dot plot: number of siblings for 12 students. Each dot is one student. The stack above 2 (four dots) shows that 2 siblings is the mode.

Worked example 2 A stem-and-leaf plot

Build a stem-and-leaf plot for: $42, 37, 51, 46, 38, 49, 52, 41, 35, 45, 50, 48$ .

Sort mentally and split each value into “tens” and “units”:

Stem | Leaf
  3  | 5 7 8
  4  | 1 2 5 6 8 9
  5  | 0 1 2

Read: $3 \mid 5 = 35$ , $4 \mid 1 = 41$ , etc.

4. Interpreting graphs

When reading any graph, ask:

What is the variable on each axis?
Which value is the largest (mode/max)? Which is the smallest?
Is there a pattern or trend?
Are there any unusual values (outliers)?

Line graph: temperature (°C) during a school day. The trend rises until 2 p.m. then falls. The peak (27 °C) is easy to read from the graph.

The line graph above shows continuous data over time. You can read the peak ( $27°$ C at $2$ p.m.), the trend (rises then falls), and the symmetry (roughly even climb and descent).

Practice

Fluency

Tier 1: basic skills

Classify as categorical, discrete numerical, or continuous numerical: eye colour.
Classify: number of pets owned.
Classify: weight of a parcel.
Classify: gender identity.
Classify: temperature at noon.
Classify: shoe size (UK sizing: 5, 5.5, 6, …).
Build a frequency table from: A, B, A, C, B, A, A, C, B, A.
A frequency table shows $5, 7, 3$ in three categories. What is the total sample size?
Which graph is best for categorical data: line graph, column graph, or stem-and-leaf?
Which graph keeps individual values visible: dot plot or column graph?
Read from the stem-and-leaf plot: $2 \mid 3\ 5\ 8$ . Write the three values.
In a dot plot, $4$ dots stack above the number $7$ . What does this mean?
A column graph has heights $8, 12, 5, 15$ . What is the sum of frequencies?
A bar graph’s vertical axis starts at $50$ instead of $0$ . Why might this be misleading?

Reasoning

Tier 2: mixed practice

Use this data set for questions 1-5: shoe sizes of $15$ students: $7, 8, 8, 9, 7, 6, 8, 10, 7, 9, 8, 9, 7, 8, 9$ .

Build a frequency table.
What is the modal shoe size (the most common)?
Describe the distribution (symmetrical, skewed, or otherwise).
If you were to draw a dot plot, how many dots would stack above $8$ ?
What type of graph would you not use for this data, and why?

The following stem-and-leaf plot shows exam marks out of $100$ for a class:

Stem | Leaf
  4  | 2 5 8
  5  | 0 3 3 7 9
  6  | 1 1 4 8
  7  | 0 2 5

How many students are in the class?
What is the lowest score? The highest score?
What mark was scored by the most students?
What is the range of the scores? (max $-$ min.)

Reasoning

Tier 3: explain and spot the mistake

Ben plots temperatures taken every hour from $6$ a.m. to $6$ p.m. as a column graph with gaps between bars. Is the column graph the best choice here? Explain.
A graph shows sales for three products with bar heights $50, 51, 52$ , and the $y$ -axis starts at $49$ . Explain why this graph could mislead a reader.
Can a single data point be both an outlier and the mode? Explain.
A friend says “categorical data can be averaged”. Is this correct? Give an example that supports your view.

Problem solving

Tier 4: real-world problems

A class survey of favourite sports gave: AFL $9$ , Soccer $7$ , Basketball $5$ , Cricket $4$ , Other $2$ . How many students were surveyed? Draw (describe) a column graph for this data.
In one week a shop recorded daily customer numbers: Mon $42$ , Tue $38$ , Wed $45$ , Thu $50$ , Fri $65$ , Sat $80$ , Sun $60$ . Which graph type would you use? What total was served?
The temperatures in a city ( degC) every hour from $9$ a.m. to $5$ p.m. were: $18, 20, 22, 24, 26, 27, 26, 24, 22$ . Which display is best? At what time was the maximum reached?
A class measured heights (cm) of $14$ students: $145, 150, 152, 150, 155, 148, 162, 158, 150, 155, 160, 153, 149, 156$ . Construct a stem-and-leaf plot.
A town’s population over $5$ decades was $12\,000$ , $15\,000$ , $22\,000$ , $28\,000$ , $31\,000$ . Which graph shows the trend best, and why?